Editing Fundamental group (section)

=== Relationship to first homology group ===
The [[abelianization]] of the fundamental group can be identified with the first [[homology group]] of the space.

A special case of the [[Hurewicz theorem]] asserts that the first [[singular homology|singular homology group]] <math>H_1(X)</math> is, colloquially speaking, the closest approximation to the fundamental group by means of an abelian group. In more detail, mapping the homotopy class of each loop to the homology class of the loop gives a [[group homomorphism]]
:<math>\pi_1(X) \to H_1(X)</math>
from the fundamental group of a topological space ''X'' to its first singular homology group <math>H_1(X).</math> This homomorphism is not in general an isomorphism since the fundamental group may be non-abelian, but the homology group is, by definition, always abelian. This difference is, however, the only one: if ''X'' is path-connected, this homomorphism is [[surjective]] and its [[Kernel (algebra)|kernel]] is the [[commutator subgroup]] of the fundamental group, so that <math>H_1(X)</math> is isomorphic to the [[abelianization]] of the fundamental group.<ref>{{harvtxt|Fulton|1995|loc=Prop. 12.22}}</ref>